Fungrim entry: dfbcd9

Symbol: Exp —

{e}^{z}

— Exponential function

The exponential function

{e}^{z}

is a function of one complex variable

z

. It can be defined by the Taylor series 1635f5.

In rendered formulas, Exp(z) is shown as

{e}^{z}

or as

\exp\!\left(z\right)

depending on the typographical requirements; no semantic difference is implied.

The following table lists all conditions such that Exp(z) is defined in Fungrim.

Domain	Codomain
Numbers
$z \in \left\{0\right\}$	${e}^{z} \in \left\{1\right\}$
$z \in \mathbb{R}$	${e}^{z} \in \left(0, \infty\right)$
$z \in \mathbb{C}$	${e}^{z} \in \mathbb{C} \setminus \left\{0\right\}$
Infinities
$z \in \left\{\infty\right\}$	${e}^{z} \in \left\{\infty\right\}$
$z \in \left\{-\infty\right\}$	${e}^{z} \in \left\{0\right\}$
Formal power series
$z \in \mathbb{Q}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] z = 0$	${e}^{z} \in \mathbb{Q}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] {e}^{z} = 1$
$z \in \mathbb{R}[[x]]$	${e}^{z} \in \mathbb{R}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] {e}^{z} \ne 0$
$z \in \mathbb{C}[[x]]$	${e}^{z} \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, [{x}^{0}] {e}^{z} \ne 0$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`RR`	$\mathbb{R}$	Real numbers
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`FormalPowerSeries`	$K[[x]]$	Formal power series
`QQ`	$\mathbb{Q}$	Rational numbers

Source code for this entry:

Entry(ID("dfbcd9"),
    SymbolDefinition(Exp, Exp(z), "Exponential function"),
    Description("The exponential function", Exp(z), "is a function of one complex variable", z, ".", "It can be defined by the Taylor series", EntryReference("1635f5"), "."),
    Description("In rendered formulas,", SourceForm(Exp(z)), "is shown as", Exp(z), "or as", Call(Exp, z), "depending on the typographical requirements; no semantic difference is implied."),
    Description("The following table lists all conditions such that", SourceForm(Exp(z)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(Element(z, Set(0)), Element(Exp(z), Set(1))), Tuple(Element(z, RR), Element(Exp(z), OpenInterval(0, Infinity))), Tuple(Element(z, CC), Element(Exp(z), SetMinus(CC, Set(0)))), TableSection("Infinities"), Tuple(Element(z, Set(Infinity)), Element(Exp(z), Set(Infinity))), Tuple(Element(z, Set(Neg(Infinity))), Element(Exp(z), Set(0))), TableSection("Formal power series"), Tuple(And(Element(z, FormalPowerSeries(QQ, x)), Equal(SeriesCoefficient(z, x, 0), 0)), And(Element(Exp(z), FormalPowerSeries(QQ, x)), Equal(SeriesCoefficient(Exp(z), x, 0), 1))), Tuple(Element(z, FormalPowerSeries(RR, x)), And(Element(Exp(z), FormalPowerSeries(RR, x)), Unequal(SeriesCoefficient(Exp(z), x, 0), 0))), Tuple(Element(z, FormalPowerSeries(CC, x)), And(Element(Exp(z), FormalPowerSeries(CC, x)), Unequal(SeriesCoefficient(Exp(z), x, 0), 0))))))

Topics using this entry

Exponential function